VARIOUS TOOLS FOR COMPARATIVE STATICS (ROUGH) 1. THE CHAIN RULE(OR TOTAL DERIVATIVE) FOR COMPOSITE FUNCTIONS OF SEVERAL VARIABLES 1.1.Chainruleforfunctionsoftwovariables.Whenyf(x 1,x 2 )withx 1 g(t)andx 2 h(t),then dy dt dx 1 dt + dx 2 dt dg(t) dt + dh(t) dt (1) Thisisusuallycalledthetotalderivativeofywithrespecttot. 1.2.Example.Letthefunctionbegivenby with y f (x 1,x 2 ) x 2 1 +x3 2 x 1 (t) t 2 +2t +1 x 2 (t) 3t Then dy x 2x 1 1, 3x2 2 dx 1 dx dt 2t +2, 2 dt 3 dt (2x 1 ) (2t +2) + (3x2 2) (3) (2t 2 +4t +2) (2t +2) + (27t 2 ) (3) 4t 3 +4t 2 +8t 2 +8t +4t +4+81t 2 4t 3 +93t 2 +12t +4 (2) Date: February 19, 2008. 1
2 VARIOUS TOOLS FOR COMPARATIVE STATICS(ROUGH) Ifwemultiplyifoutweobtain y f (x 1,x 2 ) x 2 1 +x3 2 (t 2 +2t +1) 2 + (3t) 3 t 4 +2t 3 +t 2 +2t 3 +4t 2 +2t +t 2 +2t +1+27t 3 df dt 4t 3 +6t 2 +2t +6t 2 +8t +2+2t +2+81t 2 4t 3 +93t 2 +12t +4 (3) In-class exercises Findthetotalderivativeofeachofthefollowingwithrespecttot. y f (x 1,x 2 ) x1 2 +x3 2 1. x 1 (t) t 2 x 2 (t) 2t 2. y f (x 1,x 2 ) x 2 1 +x2 2 x 1 (t) t 2 +2t x 2 (t) 2t +1 3. y f (x 1,x 2 ) x 2 1 x 1x 2 +x 2 2 x 1 (t) t 2 +2t +3 x 2 (t) 2t t 2 4. y f (x 1,x 2,x 3 ) x 2 1 +x2 2 +2x2 3 x 1 (t) t 2 +2t x 2 (t) 2t x 3 (t) t 2 5t 5. y f (x 1,x 2 ) x2 1 +x2 2 x 1 +x 2 x 1 (t) t 2 +2t x 2 (t) 2t +1 2. DIRECTIONAL DERIVATIVES 2.1.Idea.Ifyf(x 1,x 2 ),thepartialderivatives, x measuretheratesofchangeoff(x 2 1,x 2 ),in thedirectionsofthex 1 -axisandthex 2 -axis,respectively.wecanalsomeasuretherateofchange ofthefunctioninotherdirections. Consideraparticularpointinthedomainoffanddenoteit (x1 0,x0 2 ).Anynon-zerovector(h,k)isthenadirectioninwhichwemoveawayfromthepoint (x1 0,x0 2 )inastraightlinetopointsoftheform (x 1,x 2 ) (x 1 (t),x 2 (t)) (x1 0 +th,x0 2 +tk) (4) Givenanyinitialpoint (x1 0,x0 2 )andanydirection(h,k),definethedirectionalfunctiongby The derivative of this function is g (t) f (x 0 1 +th,x0 2 +tk) (5)
VARIOUS TOOLS FOR COMPARATIVE STATICS (ROUGH) 3 dg dt dx 1 dt + dx 2 dt x (x 0 1 1 +th,x0 2 +tk)h + x (x 0 2 1 +th,x0 2 +tk)k (6) Nowlett0sothatweareatthepoint (x 0 1,x0 2 ).Thenweobtain dg dt (0) x (x 0 1 1,x0 2 )h + x (x 0 2 1,x0 2 )k (7) Ifthevector(h,k)haslength1,thederivativeoffinthedirection(h,k)iscalledthedirectional derivativeoffinthedirectionof(h,k)at (x 0 1,x0 2 ).Specifically,thedirectionalderivativeoff(x 1, x 2 )at (x 0 1,x0 2 )inthedirectionoftheunitvector(h,k)is D h,k f (x1 0,x0 2 ) (x 0 1 x,x0 2 )h + (x1 0 1 x,x0 2 )k (8) 2 Notethatwhenthelengthof(h,k)isone,amoveawayfrom (x1 0, x0 2 )inthedirection(h,k) changesthevalueoffbyapproximatelyd h,k f (x1 0,x0 2 ).Alsonoticethatthedirectionalderivative istheproductofthegradientoffandthevector(h,k).
4 VARIOUS TOOLS FOR COMPARATIVE STATICS(ROUGH) 2.2.Example.Considerthefunctionf(x 1,x 2 )withthefollowingdirectionandinitialpoint. f (x 1,x 2 ) x 2 1 +3x2 2 Direction (2,5) Point (1,1) Firstnormalizethedirectionvector.Becausethelengthofthevectoris ( ) 29wecannormalizeit as 2 5,.Thenfindthegradientoffas 29 29 Evaluated at(1,1) we obtain f (x 1,x 2 ) x 2 1 +3x2 2 2x 1 6x 2 2 6 The directional derivative is then given by ( 2 ), (2,6 ) 29 5 29 34 29
VARIOUS TOOLS FOR COMPARATIVE STATICS (ROUGH) 5 3. MORE GENERAL CHAIN RULES 3.1.Generalformofthechainrule.Letyf(x 1,x 2,...x n )andletx 1 g 1 (t 1,t 2,...,t m ),x 2 g 2 (t 1, t 2,...,t m ),...,x n g n (t 1,t 2,...,t m )wheretisanm-vectorofothervariablesuponwhichthexvector depends. Then the following holds y y + y + + y x n,j 1,2,,n (9) t j t j t j x n t j 3.2.Example.Considerthefunctionyf(x 1,x 2 )alongwiththeauxiliaryfunctionsx 1 (z,w)andx 2 (z,w) y f (x 1,x 2 ) 3x 1 +2x 1 x 2 2 x 1 (z,w) 5z +2zw x 2 (z,w) zw 2 +3w wheret 1 zandt 2 wfromequation9.wecanfindthepartialderivativeofywithrespectto zusingequation9asfollows. y z (x 1,x 2 ) z + (x 1,x 2 ) z (3 +2x 2 2 ) (5 +2w) + (4x 1x 2 ) (w 2 ) (3 +2 (z 2 w 4 +6zw 3 +9w 2 ) ) (5 +2w) +4 (5z +2zw) (zw 2 +3w )w 2 (3 +2z 2 w 4 +12zw 3 +18w 2 ) (5 +2w) + (20zw 2 +8zw 3 ) (zw 2 +3w ) 15 +6w +10z 2 w 4 +4z 2 w 5 +60zw 3 +24zw 4 +90w 2 +36w 3 +20z 2 w 4 +60zw 3 +8z 2 w 5 +24zw 4 15 +6w +90w 2 +36w 3 +120zw 3 +48zw 4 +30z 2 w 4 +12z 2 w 5 Wecanalsofindthepartialofywithrespecttowas y w (x 1,x 2 ) w + (x 1,x 2 ) w (3 +2x 2 2 ) (2z) + (4x 1x 2 ) (2zw +3) (3 +2 (z 2 w 4 +6zw 3 +9w 2 ) ) (2z) +4 (5z +2zw) (zw 2 +3w ) (2zw +3) (3 +2z 2 w 4 +12zw 3 +18w 2 ) (2z) + (20z +8zw) (zw 2 +3w ) (2zw +3) (6z +4z 3 w 4 +24z 2 w 3 +36zw 2 ) + (20z 2 w 2 +60zw +8z 2 w 3 +24zw 2 ) (2zw +3) 6z +4z 3 w 4 +24z 2 w 3 +36zw 2 +40z 3 w 3 +120z 2 w 2 +16z 3 w 4 +48z 2 w 3 +60z 2 w 2 +180zw +24z 2 w 3 +72zw 2 6z +180zw +108zw 2 +180z 2 w 2 +96z 2 w 3 +40z 3 w 3 +20z 3 w 4
6 VARIOUS TOOLS FOR COMPARATIVE STATICS(ROUGH) 4.1. A two-variable implicit function theorem. 4. DERIVATIVES OF IMPLICIT FUNCTIONS 4.1.1.statementofthetheorem.Considerthefunctiondefinedimplicitlybyf(x 1,x 2 )cwherecisa constant.if 0,then 4.1.2. examples. a:example1 (10) y 0 f (x 1,x 2 ) x 2 1 2x 1x 2 +x 1 x 3 2 Tofindthepartialderivativeofx 2 withrespecttox 1 wefindthetwopartialsoffas follows x 1 2x 1 2x 2 +x2 3 3x 1 x2 2 2x 1 2x 2 2x 1 x 3 2 3x 1 x 2 2 2x 1 b:example2 u 0 U (x 1,x 2 ) Tofindthepartialderivativeofx 2 withrespecttox 1 wefindthetwopartialsofuas follows U U 4.2. A more general form of the implicit function theorem with p independent variables and 1 implicit equation. 4.2.1.Statementofthetheorem.Supposex 1 isdefinedimplicitlyasadifferentiablefunctionofthep variablesy 1,y 2,...,y p bytheequationf(x 1,y 1,y 2,...y p )0orf(x 1,y 1,y 2,...y p )cwherecis a constant. Then y i y i,i 1,2,,p (11)
VARIOUS TOOLS FOR COMPARATIVE STATICS (ROUGH) 7 4.2.2. Example. Consider a utility function given by This can be written implicitly as u x 1 4 1 x 1 3 2 x 1 6 3 f (u,x 1,x 2,x 3 ) u x 1 4 1 x 1 3 2 x 1 6 3 0 Now consider some of the partial derivatives arising from this implicit function. First consider themarginalrateofsubstitutionofx 1 forx 2 1 ( 1 3 )x 41 x 2 1 3 2 x6 3 ( 4 1 1 1 )x 3 4 1 x3 2 x 63 3 4 x 1 x 2 Nowconsiderthemarginalrateofsubstitutionofx 2 forx 3. x 3 Nowconsiderthemarginalutilityofx 2. u x 3 1 1 ( 1 6 )x 41 x3 2 x 5 6 3 ( 3 1 1 )x 41 x 2 1 3 2 x6 3 1 2 u x 2 x 3 ( 1 3 )x 1 41 x 2 3 2 x1 63 1 1 3 x1 4 1 x 2 3 2 x 1 6 3
8 VARIOUS TOOLS FOR COMPARATIVE STATICS(ROUGH) 4.3. A more general form of the implicit function theorem with p independent variables and m implicit equations. 4.3.1. One motivation for implicit function theorem. Assuming the conditions of the implicit function theorem as discussed below hold, we can eliminate m variables from the constrained optimization problem using the constraint equations. In this way the constrained problem is converted to an unconstrained problem and we can use the results on unconstrained problems to determine a solution.. 4.3.2.descriptionofthesystemofequations. Supposethatwehavemequationsdependingonm+p variables(parameters) written in implicit form as follows φ 1 (x 1,x 2,,x m, y 1,y 2,,y p ) 0 φ 2 (x 1,x 2,,x m, y 1,y 2,,y p ) 0 0........ φ m (x 1,x 2,,x m, y 1,y 2,,y p ) 0 (12) Forexamplewithm2andp3,wemighthave φ 1 (x 1,x 2,p,w 1,w 2 ) (0.4)px1 0.6 x2 0.2 w 1 0 φ 2 (x 1,x 2,p,w 1,w 2 ) (0.2)px1 0.4 x2 0.8 w 2 0 (13) 4.3.3.Jacobianmatrixofthesystem.TheJacobianmatrixofthesystemin(12)definedasmatrixof first partials as follows J φ 1 φ 1 φ 2 φ 2...... φ m φ m φ 1 x m φ 2 x m.. φ m x m (14) Thismatrixwillbeofrankmifitsdeterminantisnotzero. 4.4.Statementofimplicitfunctiontheorem.Suppose φ i arerealvaluedfunctionsdefinedona domaindandarecontinuouslydifferentiableonanopensetd 1 D R m+p,p 0and φ i (x 0,y 0 ) 0fori1,2,...,mand(x 0,y 0 ) ǫd 1,i.e.,theequationsystemissatisfiedatthepoint(x 0,y 0 ).Letx beofdimensionmandyofdimensionp.assumethejacobianmatrix [ Œi (x 0,y 0 ) x j hasrankm.thenthereexistsaneighborhoodof(x 0,y 0 )(N ffi (x 0,y 0 )) ǫd 1,anopensetD 2 R p containingy 0,andrealvaluedfunctions ψ k k1,...,mcontinuouslydifferentiableond 2 suchthat the following conditions are satisfied x 0 k g k (y 0 )k 1,...,m. (15) Thismeanswecansolvethesystemofmimplicitequationsforallmofthexvariablesasfunctionsofthepindependentoryvariables. Intheexamplesystem(13),wecouldsolvethetwo
VARIOUS TOOLS FOR COMPARATIVE STATICS (ROUGH) 9 equationsforx 1 andx 2 asfunctionsofp,w 1,andw 2,i.e.,x 1 g 1 (p,w 1,w 2 )andx 2 g 2 (p,w 1,w 2 ). Foreveryy ǫd 2,wehave φ i (g 1 (y), g 2 (y),g m (y),y) 0i 1,...,m φ i (g (y),y) 0i 1,...,m (16) whereg(y)[g 1 (y),g 2 (y)...g m (y)andforall(x,y) ǫn ffi (x 0,y 0 )thejacobianmatrix φ i (x,y) x j hasrankm (17) Thismeansthatifweplugthex sthatweobtainfrom(16)intotheimplicitequations(equation 12),thelefthandsideoftheequationswillbezero,ortheequationswillbesatisfied. FurthermoreforyǫD 2,wecancomputethepartialderivativesof ψ k (y)assolutionstothesetof equations m φ i (g (y),y) x k1 k Herethederivative φ i(g(y),y) g k (y) y j φ i(g (y),y) y j,i 1,2,,m (18) x isjustthederivativeof φ k i withrespecttoitskthargument. Inequation12thisisx ( ) k,whileinequation16,itisg k (y). Fortheexamplein(13)wecanfind,thatis gi, from the following two equations derived from equation 13, which is repeated here. φ 1 (x 1,x 2,p,w 1,w 2 ) (0.4)px1 0.6 x2 0.2 w 1 0 φ 2 (x 1,x 2,p,w 1,w 2 ) (0.2)px1 0.4 x2 0.8 w 2 0 [ [ ( 0.24)px1 1.6 x2 0.2 x1 + (0.08)px1 0.6 x2 0.8 x2 (0.4)x1 0.6 x2 0.2 [ [ (0.08)px1 0.6 x2 0.8 x1 + ( 0.16)px1 0.4 x2 1.8 x2 (0.2)x1 0.4 x2 0.8 (19) While the system is non-linear, we can(in principle and in practice) solve for the two derivatives in question. Toseetheintuitionofequation18,takethetotalderivativeof φ i in(16)withrespecttoy j as follows φ i (g 1 (y), g 2 (y),, g m (y),y) 0 φ i g 1 g 1 y + φ i g 2 j g 2 y + + φ i g m j g m y + φ i j y 0 j (20) andthenmove φ i y j totherighthandsideoftheequation.thenperformasimilartaskforthe otherequationstoobtainmequationsinthempartialderivatives, x i y j g i y j Forthecaseofonlyoneimplicitequation(18)reducesto m φ i (g (y),y) g k (y) φ(ψ(y),y) (21) x k1 k y j y j Withonlyoneimplicitequationm1andweobtain
10 VARIOUS TOOLS FOR COMPARATIVE STATICS(ROUGH) φ(g (y),y) x k g k (y) φ (g (y),y) (22) y j y j which can be rewritten as g k (y) y j φ(g(y),y) y j φ(g(y),y) x k (23) Thisisthenthesameasequation11. Ifthereareonlytwovariables,x 1 andx 2 wherex 2 isnowlikey 1,weobtain φ(g(x 2 ),x 2 ) g 1 (x 2 ) φ(g (x 2),x 2 ) g 1(x 2 ) φ (g (x 2 ),x 2 ) φ(g(x 2 ),x 2 ) (24) whichisthesameasthetwovariablecaseinequation10where φtakestheplaceoff. 4.4.1. examples of implicit function theorem. a:example1 TheJacobianisgivenby φ 1 (x 1,x 2,y) 3x 1 +2x 2 +4y 0 φ 2 (x 1,x 2,y) 4x 1 +x 2 +y 0 [ 3 2 4 1 SincethisJacobianisoffullrank,wecansolvetheseequationsforx 1 andx 2 asfunctions ofy.moveytotherhsineachequation. 3x 1 +2x 2 4y 4x 1 +x 2 y Nowsolvethesecondequationforx 2,andpluginthefirstequation
VARIOUS TOOLS FOR COMPARATIVE STATICS (ROUGH) 11 x 2 y 4x 1 3x 1 +2( y 4x 1 ) 4y 3x 1 2y 8x 1 4y 5x 1 2y x 1 2/5y ψ 1 (y) Nowplugthisanswerintotheequationfory 2 andsolve x 2 y 4(2/5y) y 8/5y 13/5y ψ 2 (y) Wecanchecktoseethatequation16issatisfied Furthermore φ 1 (2/5y, 13/5y,y) 3(2/5y) +2 ( 13/5) +4y 6/5y 26/5y +4y 0 φ 2 (2/5y, 13/5y,y) 4(2/5y) + ( 13/5) +y 8/5y 13/5y +y 0 y g 1(y) y 2 5 y g 2(y) y 13 5
12 VARIOUS TOOLS FOR COMPARATIVE STATICS(ROUGH) Ratherthansolvingthesystemforx ( ) ( ) 1 andx 2 asfunctionsofy,wecansolveforthepartial derivatives x1 g1 g 2 y y y y usingtheformulain(18)as: φ 1 g 1 y + φ 1 g 2 y φ 1 y φ 2 g 1 y + φ 2 g 2 y φ 2 y 3 g 1 y +2 g 2 2y 4 4 g 1 y + g 2 y 1 > g 2 y 1 4 g 1 y > 3 g 1 y +2 ( 1 4 g 1 y ) 4 > 3 g 1 y 2 8 g 1 y 4 5 g 1 y 2 g 1 y 2/5 g 2 y 1 4 (2/5) 1 8/5 13/5
VARIOUS TOOLS FOR COMPARATIVE STATICS (ROUGH) 13 b:example2 Lettheproductionfunctionforafirmbygivenby y 14x 1 +11x 2 x 2 1 x2 2 Profitforthefirmisgivenby π py w 1 x 1 w 2 x 2 p (14x 1 +11x 2 x 2 1 x2 2 ) w 1x 1 w 2 x 2 The first order conditions for profit maximization imply that π 14px 1 +11px 2 px1 2 px2 2 w 1x 1 w 2 x 2 π φ 1 14p 2px 1 w 1 0 π φ 1 11p 2px 2 w 2 0 Wecansolvethefirstequationforx 1 asfollows π 14p 2px 1 w 1 0 2px 1 14p w 1 x 1 14p w 1 2p 7 w 1 2p Inasimilarmannerwecanfindx 2 fromthesecondequation π 11p 2px 2 w 2 0 2px 2 11p w 2 x 2 11p w 2 2p 5.5 w 2 2p
14 VARIOUS TOOLS FOR COMPARATIVE STATICS(ROUGH) Wecanfindthederivativesofx 1 andx 2 withrespecttop,w 1 andw 2 directlyasfollows: x 1 7 1 2 w 1p 1 x 2 5.5 1 2 w 2p 1 1 2 w 1p 2 w 1 1 2 p 1 w 2 0 1 2 w 2p 2 w 2 1 2 p 1 w 2 0 We can also find these derivatives using the implicit function theorem. The two implicit equations are φ 1 (x 1,x 2,p,w 1,w 2 ) 14p 2px 1 w 1 0 φ 2 (x 1,x 2 p,w 1,w 2 ) 11p 2px 2 w 2 0 FirstwechecktheJacobianofthesystem.Itisobtainedbydifferentiating φ 1 and φ 2 with respecttox 1 andx 2 asfollows J [ φ1 φ 1 φ 2 φ 2 [ 2p 0 0 2p Thedeterminantis4p 2 whichispositive. Nowwehavetwowecansolveusingthe implicit function theorem.
VARIOUS TOOLS FOR COMPARATIVE STATICS (ROUGH) 15 The theorem says m k1 φ i (g (y),y) x k or φ 1 + φ 1 φ 1 φ 2 + φ 2 φ 2 [ φ1 φ 1 φ 2 φ 2 [ x1 [ x1 g k (y) y j φ i(ψ(y),y) y j,i 1,2, [ φ 1 φ 2 [ φ1 φ 1 φ 2 φ 2 1 [ φ 1 φ 2 We can compute the various partial derivatives of φ as follows (25) φ 1 (x 1,x 2,p,w 1,w 2 ) 14p 2px 1 w 1 0 φ 1 2p φ 1 0 φ 1 14 2x 1 φ 2 (x 1,x 2 p,w 1,w 2 ) 11p 2px 2 w 2 0 φ 2 2p φ 2 0 φ 1 11 2x 2
16 VARIOUS TOOLS FOR COMPARATIVE STATICS(ROUGH) Nowwritingoutthesystemweobtainforthecaseathandweobtain [ x1 [ x1 [ [ φ1 φ 1 φ 2 φ 2 [ 2p 0 0 2p [ 1 2p 0 0 1 7 x 1 p 5.5 x 2 p 2p 1 [ φ 1 φ 2 1 [ 2x1 14 2x 2 11 [ 2x1 14 2x 2 11 (26) Given that x 1 7 1 2 w 1p 1 x 2 5.5 1 2 w 2p 1
VARIOUS TOOLS FOR COMPARATIVE STATICS (ROUGH) 17 we obtain 7 x 1 p 7 (7 2 1w 1p 1 ) p 2 1w 1p 2 5.5 x 2 p 5.5 (5.5 2 1w 2p 1 ) p 2 1w 2p 2 whichisthesameasbefore. c: example 3(exercise) Verifytheimplicitfunctiontheoremderivatives w 1, w 2, w 1, w 2 forthefunctionin example 2. d:example4 Afirmsellsitsoutputintoaperfectlycompetitivemarketandfacesafixedpricep. It hireslaborinacompetitivelabormarketatawagew,andrentscapitalinacompetitive capital market at rental rate r. The production is f(l, K). The production function is strictly concave. The firm seeks to maximize its profits which are ß pf(l,k) wl rk The first-order conditions for profit maximization are π L pf L (L,K ) w 0 π K pf K (L,K ) r 0 ThisgivestwoimplicitequationsforKandL.
18 VARIOUS TOOLS FOR COMPARATIVE STATICS(ROUGH) The second order conditions are 2 π L 2 (L, K ) < 0 2 π 2 π L 2 K 2 We can compute these derivatives as [ 2 2 π > 0at (L, K ) L K 2 π (pf L(L,K ) w ) L 2 L 2 π (pf K(L,K ) r ) K 2 L 2 π L K (pf L(L,K ) w ) K pf LL pf KK pf KL Thefirstofthesecondorderconditionsissatisfiedbyconcavityoff.Wethenwritethe second condition as D pf LL pf LK pf KL pf KK > 0 p 2 f LL f LK f KL f KK > 0 (27) p 2 (f LL f KK f KL f LK ) > 0 The expression is positive because f is assumed to be strictly concave. Nowwewishtodeterminetheeffectsoninputdemands,L andk,ofchangesinthe input prices. Using the implicit function theorem in finding the partial derivatives with respecttow,weobtainforthefirstequation π LL L For the second equation we obtain w + π LK K w + π Lw 0 L pf LL w + pf KL K w 1 0 π KL L w + π KK K w + π Kw 0 L pf KL w + pf KK K w 0 Wecanwritethisinmatrixformas [ pfll pf LK pf KL pf KK [ L / w K / w 1 0 Using Cramer s rule, we then have the comparative-statics results L w K w 1 pf LK 0 pf KK D pf LL 1 pf KL 0 D pf KK D pf KL D Thesignofthefirstisnegativebecausef KK <0andD >0fromeitherstrictconcavity of f or the second-order conditions. Thus, the demand curve for labor has a negative slope.
VARIOUS TOOLS FOR COMPARATIVE STATICS (ROUGH) 19 However,inordertosigntheeffectofachangeinthewagerateonthedemandforcapital, weneedtoknowthesignoff KL,theeffectofachangeinthelaborinputonthemarginal productofcapital.itisplausibletoassumethatthisispositive(thoughitmaynotbe)and soanincreaseinthewageratewouldalsodecreasethedemandforcapital.wecanderive theeffectsofachangeintherentalrateofcapitalinasimilarway.
20 VARIOUS TOOLS FOR COMPARATIVE STATICS(ROUGH) e:example5 LettheproductionfunctionbeCobb-DouglasoftheformyL ff K fi.findthecomparativestaticeffects L / wand K / w. Profits are given by π pf(l,k) wl rk pl α K β wl rk The first-order conditions are for profit maximization are π L αpl α 1 K β w 0 π K βpl α K β 1 r 0 ThisgivestwoimplicitequationsforKandL. The second order conditions are 2 π L 2 (L, K ) < 0 2 π 2 π L 2 K 2 We can compute these derivatives as [ 2 2 π > 0at (L, K ) L K 2 π α (α 1)pL α 2 K β L 2 2 π β (β 1)pL α K β 2 K 2 2 π L K α βplα 1 K β 1 Thefirstofthesecondorderconditionsissatisfiedaslongas α, β 1.Wecanwritethe second condition as D α (α 1)pL α 2 K β α βpl α 1 K β 1 α βpl α 1 K β 1 β (β 1)pL α K β 2 > 0 (28)
VARIOUS TOOLS FOR COMPARATIVE STATICS (ROUGH) 21 WecansimplifyDasfollows D α (α 1)pL α 2 K β α βpl α 1 K β 1 α βpl α 1 K β 1 β (β 1)pL α K β 2 p 2 α (α 1)L α 2 K β α βl α 1 K β 1 α βl α 1 K β 1 β (β 1)L α K β 2 p 2 α β (α 1) (β 1)L α 2 K β L α K β 2 α 2 β 2 L 2α 2 K p 2 α βl α 2 K β L α K β 2 ( (α 1) (β 1) α β ) p 2 α βl α 2 K β L α K β 2 ( α β β α +1 α β ) p 2 α βl α 2 K β L α K β 2 (1 α β ) 2β 2 (29) The condition is then that D p 2 α βl 2α 2 K 2β 2 (1 α β) > 0 Thiswillbetrueif α + β <1. Nowwewishtodeterminetheeffectsoninputdemands,L andk,ofchangesinthe input prices. Using the implicit function theorem in finding the partial derivatives with respecttow,weobtainforthefirstequation π LL L w + π LK K α (α 1)pL α 2 K β L w + α βplα 1 K For the second equation we obtain w + π Lw 0 1 0 β 1 K w L π KL w + π KK K w + π Kw 0 α βpl α 1 Kβ 1 L w + β (β 1)pLα Kβ 2 K w 0 Wecanwritethisinmatrixformas [ α (α 1)pL α 2 K β α βpl α 1 K β 1 α βpl α 1 K β 1 β (β 1)pL α K β 2 [ L / w K / w 1 0
22 VARIOUS TOOLS FOR COMPARATIVE STATICS(ROUGH) Using Cramer s rule, we then have the comparative-statics results L w K w 1 α βpl α 1 K β 1 0 β (β 1)pL α K β 2 D α (α 1)pL α 2 K β 1 α βpl α 1 K β 1 0 D β (β 1)pLα K β 2 D α βplα 1 K β 1 D Thesignofthefirstofisnegativebecausep β (β 1)L α K β 2 <0(β <1),and D > 0bythesecondorderconditions.Thecrosspartialderivative K w isalsolessthanzero becausep α βl α 1 K β 1 > 0andD > 0.So,forthecaseofaCobb-Douglasproduction function, an increase in the wage unambiguously reduces the demand for capital.
VARIOUS TOOLS FOR COMPARATIVE STATICS (ROUGH) 23 5. THEGENERALEQUATIONFORTHETANGENTTOF(X 1,X 2,... )CANDPROPERTIESOFTHE GRADIENT 5.1. The general equation for a tangent plane. Consider the surface defined by atthepoint (x 0 1,x0 2,,x0 n ) y f (x 1,x 2,,x n ) (30) y 0 f (x 0 1,x0 2,,x0 n) (31) Theequationoftheplanetangenttothesurfaceatthispointisgivenby (y y 0 ) f 1 (x 1 x 0 1 ) + f 2 (x 2 x 0 2 ) + + f n (x n x 0 n) f 1 (x 1 x 0 1 ) +f 2 (x 2 x 0 2 ) + + f n (x n x 0 n) (y y 0 ) 0 f (x 1,x 2,,x n ) f (x 0 1,x0 2,,x0 n) + f 1 (x 1 x 0 1 ) + f 2 (x 2 x 0 2 ) + + f n (x n x 0 n) (32) wherethepartialderivativesf i areevaluatedat (x 0 1,x0 2,,x0 n ).Wecanalsowritethisas Compare this to the tangent equation with one variable f (x) f (x 0 ) f(x 0 ) (x x 0 ) (33) y f (x 0 ) f (x 0 ) (x x 0 ) y f (x 0 ) + f (x 0 ) (x x 0 ) (34) 5.2.Vectorfunctionsofarealvariable.Letx 1,x 2,...,x n befunctionsofavariabletdefinedinan interval I and write x (t) (x 1 (t),x 2 (t),,x n (t) ) (35) Thefunctiont x(t)isatransformationfromrtor n andiscalledavectorfunctionofareal variable.asxrunsthroughi,x(t)tracesoutasetofpointsinn-spacecalledacurve.inparticular, ifweput x 1 (t) x 0 1 +ta 1,x 2 (t) x 0 2 +ta 2,,x n (t) x 0 n +ta n (36) theresultingcurveisastraightlineinn-space.itpassesthroughthepointx 0 (x 0 1,x0 2,,x0 n ) att0,anditisinthedirectionofthevectora (a 1,a 2,,a n ). Wecandefinethederivativeofx(t)as dx dt ẋ (t) ( dx1 (t) dt, dx 2(t) dt,, dx n(t) dt IfKisacurveinn-spacetracedoutbyx(t),thenẋ (t)canbeinterpretedasavectortangenttok atthepointt. ) (37)
24 VARIOUS TOOLS FOR COMPARATIVE STATICS(ROUGH) 5.3.Tangentvectors.Considerthefunctionr (t) (x 1 (t),x 2 (t),,x n (t) )anditsderivative ṙ (t) (ẋ 1 (t), ẋ 2 (t),, ẋ n (t) ).ThisfunctiontracesoutacurveinR n.wecancallthecurve K.Forfixedt, r (t +h) r (t) ṙ (t) lim (38) h 0 h Ifṙ(t) 0,thenfort+hcloseenoughtot,thevectorr(t+h)-r(t)willnotbezero.Ashtendsto0, thequantityr(t+h)-r(t)willcomecloserandclosertoservingasadirectionvectorforthetangent tothecurveatthepointp.thiscanbeseeninthefollowinggraph Itmaybetemptingtotakethisdifferenceasapproximationofthedirectionofthetangentand then take the limit lim [r (t +h) r (t) (39) h 0 andcallthelimitthedirectionvectorforthetangent.butthelimitiszeroandthezerovector hasnodirection.insteadweusetheavectorthatforsmallhhasagreaterlength,thatisweuse r (t +h) r (t) h Foranyrealnumberh,thisvectorisparalleltor(t+h)-r(t).Thereforeitslimit r (t +h) r (t) ṙ (t) lim h 0 h canbetakenasadirectionvectorforthetangentline. 5.4. Tangent lines, level curves and gradients(sydsaeter). Consider the equation (40) (41) f (x) f (x 1,x 2,,x n ) c (42) whichdefinesalevelcurveforthefunctionf. Letx 0 (x 0 1, x0 2,, x0 n )beapointonthe surfaceandletx (t) (x 1 (t),x 2 (t),,x n (t) )representadifferentiablecurveklyingonthe surfaceandpassingthroughx 0 attt 0.BecauseKliesonthesurface,f[x(t)f[x 1 (t),x 2 (t),,,,x n (t) cforallt.nowdifferentiatethisequationwithrespecttot (x) t + (x) t + + (x) t 0 f(x 0 ) ẋ (t 0 ) 0 (43) Becausethevectorẋ (t) (ẋ 1 (t),ẋ 2 (t),,ẋ n (t) )hasthesamedirectionasthetangentto thecurvekatx 0,thegradientoffisorthogonaltothecurveKatthepointx 0.
VARIOUS TOOLS FOR COMPARATIVE STATICS (ROUGH) 25 6. LINEAR APPROXIMATIONS AND DIFFERENTIALS 6.1.Differentials.Considerafunctionyf(x 1,x 2,...,x n ).Ifdx 1,dx 2,...,dx n arearbitraryreal numbers(notnecessarilysmall),wedefinethedifferentialofyf(x 1,x 2,...,x n )as dy dx + dx 2 + + dx n (44) 1 x n Whenx i ischangedtox i +dx i,thentheactualchangeinthevalueofthefunctionistheincrement y f (x 1 +dx 1, x 2 +dx 2,, x n +dx n ) f (x 1,x 2,,x n ) (45) Ifdx i issmallinabsolutevalue,the ycanbeapproximatedbydy y dy dx 1 + dx 2 + + x n dx n (46) 6.2. Rules for differentials. 1:dc0(cisaconstant) 2:d(cx n )cnx n 1 dx 3: d (af +bg) adf +bdg(aandbareconstants) 4: d (fg) gdf + fdg 5: d ( f g ) gdf fdg g 2 g 0 6: d (fgh) ghdf + fhdg + fgdh 7:Ifyg[f(x 1,x 2,...,x n )thendyg [f(x 1,x 2,...,x n )df.
26 VARIOUS TOOLS FOR COMPARATIVE STATICS(ROUGH) 6.3. Differentials and systems of equations. 6.3.1. Idea. We can find partial derivatives of implicit systems using differentials. We take the total differential of both sides of each equation, set all differentials of variables that are not changing equal to zero, and then divide each equation by the differential of the one exogenous variable that is changing. We then solve the resulting system for the various partial derivatives. 6.3.2. Example 1. Consider the system φ 1 (x 1,x 2,p,w 1,w 2 ) 14p 2px 1 w 1 0 φ 2 (x 1,x 2 p,w 1,w 2 ) 11p 2px 2 w 2 0 The total differential of each equation is (47) 14dp 2pdx 1 2x 1 dp dw 1 0 11dp 2pdx 2 2x 2 dp dw 2 0 Nowsetdw 1 dw 2 0anddivideeachequationbydp Solving we obtain 14 2p dx 1 dp 2x 1 0 11 2p dx 2 dp 2x 2 0 2p 14 2x 1 14 2x 1 2p 7 x 1 p 2p 11 2x 1 11 2x 1 2p 5.5 x 2 p
VARIOUS TOOLS FOR COMPARATIVE STATICS (ROUGH) 27 6.3.3. Example 2. Consider the following macroeconomic model: Y C + I +G C f (Y T) I h (r) r m (M) (48) WhereYisnationalincome,Cisconsumption,Iisinvestment,Gispublicexpenditure,Tistax revenue,ristheinterestrate,andmismoneysupply.therearesevenvariablesandfourequations sowecanpotentiallysolvefor4endogenousvariablesintermsof3exogenousvariables. Ifwe assumethatf,h,andmaredifferentiablefunctionswith0 <f <1,h <0,andm <0,thenthese equationsdeterminey,c,i,andrasdifferentiablefunctionsofm,t,andg.wecanalsofindthe differentialsofy,c,i,andrintermsofthedifferentialsofm,t,andg. The total differential of the system is dy dc +di +dg dc f (Y T) (dy dt) di h (r)dr dr m (M)dM (49) WeneedtosolvethissystemforthedifferentialchangesdY,dC,dI,anddrintermsofthe differentialchangesdm,dt,anddgintheexogenouspolicyvariablesm,t,andg.fromthelast twoequationsin(41),wecanfinddianddrasfollows dr m (M)dM di h (r)m (M)dM (50) InsertingtheexpressionfordIfrom(50)intothefirsttwoequationsin(49)gives dy dc h (r)m (M)dM +dg f (Y T)dY dc f (Y T)dT (51) ThisgivestwoequationstodeterminethetwounknownsdYanddCintermsofdM,dG,and dt.wecanwritethisinmatrixformasfollows [ 1 1 f (Y T) 1 [ dy dc [ h (r)m (M)dM +dg f (Y T)dT (52) WecanuseCramer sruletosolvethissystem.thedeterminantofthecoefficientmatrixisgiven by D 1 1 f (Y T) 1 ( 1) ( f (Y T) ) f (Y T) 1 (53) FirstsolvingfordYweobtain
28 VARIOUS TOOLS FOR COMPARATIVE STATICS(ROUGH) dy dy h (r)m (M)dM +dg 1 f (Y T)dT 1 1 1 f (Y T) 1 h (r)m (M)dM dg +f (Y T)dT f (Y T) 1 h (r)m (M) f (Y T) 1 dm 1 f (Y T) 1 dg + h m 1 f dm f 1 f dt + 1 1 f dg ThensolvingfordCweobtain h (r)m (M)dM +dg 1 f (Y T)dT 1 f (Y T) 1 f (Y T) f (Y T) 1 dt (54) dc 1 h (r)m (M)dM +dg f (Y T) f (Y T)dT 1 1 f (Y T) 1 1 h (r)m (M)dM +dg f (Y T) f (Y T)dT f (Y T) 1 dc f (Y T)dT h (r)m (M)dMf (Y T) dgf (Y T) f (Y T) 1 h (r)m (M)f (Y T) f (Y T) 1 dm f (Y T) f (Y T) 1 dg + f (Y T) f (Y T) 1 dt f h m 1 f dm f 1 f dt + f 1 f dg (55) WehavenowfoundthedifferentialsdY,dC,dI,anddraslinearfunctionsofdM,dT,anddG.If wesetdmanddgequaltozero,then dy f 1 f dt Y T f 1 f Similarly T r I 0and T 0.Becauseweassumedthat0 <f <1, Y T f (1 f ) <0. IfdM,dT,anddGaresmallinabsolutevalue,then (56) Y Y(M 0 +dm,t 0 +dt,g 0 +dg) Y(M 0,T 0,G 0 ) dy
VARIOUS TOOLS FOR COMPARATIVE STATICS (ROUGH) 29 Literature Cited Sydsaeter, Knut. Topics in Mathematical Analysis for Economists. New York: Academic Press, 1981.